Find the exact value of cos(2θ)and sin(2θ) if tan(θ) = (-5/12) and 270° < θ < 360
Use double angle identities:
sin 2θ = 2 tan θ / ( 1 + tan² θ )
cos 2θ = ( 1 - tan² θ ) / ( 1 + tan² θ )
In this case:
sin 2θ = 2 ∙ ( - 5 / 12 ) / [ 1 + ( - 5 / 12 )² ] =
- 2 ∙ 5 / 12 / ( 1 + 25 / 144 ) = - 10 / 12 / ( 144 / 144 + 25 / 144 ) =
- 10 / 12 / 169 / 144 = - 10 ∙ 144 / 12 ∙ 169 =
- 10 ∙ 12 ∙ 12 / 12 ∙ 169 = - 10 ∙ 12 / 169 = - 120 / 169
cos 2θ = ( 1 - tan² θ ) / ( 1 + tan² θ )
cos 2θ = [ 1 - ( - 5 / 12 )² ] / [ 1 + ( - 5 / 12 )² ] =
( 1 - 25 / 144 ) / ( 1 + 25 / 144 ) =
( 144 / 144 - 25 / 144 ) / ( 144 / 144 + 25 / 144 ) =
119 / 144 / 169 / 144 = 144 ∙ 119 / 144 ∙ 169 = 119 / 169
tanθ = -5/12 and θ is in the fourth quadrant
tanθ = y/x and in IV, x = 12 and y = -5
r^2 = x^2 + y^2 = 144+25 = 169
So r = 13
sinθ = -5/13 and cosθ = 12/13
sin 2θ = 2sinθcosθ = 2(-5/13)(12/13) = -120/169
cos 2θ = cos^2 θ - sin2 θ = 144/169 - 25/169 = 119/169
To find the exact values of cos(2θ) and sin(2θ), we need to use trigonometric identities.
We are given that tan(θ) = -5/12 and 270° < θ < 360°.
Since tan(θ) = opposite/adjacent, we can assign a right triangle using the given information.
Let's label the opposite side as -5 and the adjacent side as 12.
Using the Pythagorean theorem, we can find the hypotenuse (h):
h^2 = (-5)^2 + 12^2
h^2 = 25 + 144
h^2 = 169
h = √169
h = 13
Now that we have the triangle, we can find the values of cosine and sine.
cos(θ) = adjacent/hypotenuse = 12/13
sin(θ) = opposite/hypotenuse = -5/13
To find cos(2θ) and sin(2θ), we can use the double-angle identities:
cos(2θ) = cos^2(θ) - sin^2(θ)
sin(2θ) = 2cos(θ)sin(θ)
Plugging in the values we found earlier:
cos(2θ) = (12/13)^2 - (-5/13)^2
= (144/169) - (25/169)
= 119/169
sin(2θ) = 2(12/13)(-5/13)
= -120/169
Therefore, the exact values are:
cos(2θ) = 119/169
sin(2θ) = -120/169
To find the exact value of cos(2θ) and sin(2θ), we need to first find the value of cos(θ) and sin(θ) based on the given information.
Since tan(θ) = (-5/12), we can use the definition of tangent to find the other trigonometric functions:
tan(θ) = opposite / adjacent
In this case, the opposite side is -5 and the adjacent side is 12. We can use the Pythagorean theorem to find the hypotenuse:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = (-5)^2 + 12^2
hypotenuse^2 = 25 + 144
hypotenuse^2 = 169
Taking the square root of both sides, we get:
hypotenuse = √169
hypotenuse = 13
Now we can find the values of cos(θ) and sin(θ) using the definitions:
cos(θ) = adjacent / hypotenuse
cos(θ) = 12 / 13
sin(θ) = opposite / hypotenuse
sin(θ) = -5 / 13
Now we can use the double-angle formulas to find the values of cos(2θ) and sin(2θ):
cos(2θ) = cos^2(θ) - sin^2(θ)
cos(2θ) = (cos(θ))^2 - (sin(θ))^2
cos(2θ) = (12/13)^2 - (-5/13)^2
cos(2θ) = 144/169 - 25/169
cos(2θ) = 119/169
sin(2θ) = 2 * sin(θ) * cos(θ)
sin(2θ) = 2 * (-5/13) * (12/13)
sin(2θ) = -120/169
Therefore, the exact value of cos(2θ) = 119/169 and the exact value of sin(2θ) = -120/169 when 270° < θ < 360° and tan(θ) = -5/12.